3.38.8 \(\int -\frac {9 e^5}{3 e^5 x+2 x^2} \, dx\)

Optimal. Leaf size=19 \[ 3 \log \left (\frac {5}{4} \left (-1-\frac {3 e^5}{2 x}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 615} \begin {gather*} 3 \log \left (2 x+3 e^5\right )-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9*E^5)/(3*E^5*x + 2*x^2),x]

[Out]

-3*Log[x] + 3*Log[3*E^5 + 2*x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 615

Int[((b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[Log[x]/b, x] - Simp[Log[RemoveContent[b + c*x, x]]/b,
x] /; FreeQ[{b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (9 e^5\right ) \int \frac {1}{3 e^5 x+2 x^2} \, dx\right )\\ &=-3 \log (x)+3 \log \left (3 e^5+2 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.68 \begin {gather*} -9 e^5 \left (\frac {\log (x)}{3 e^5}-\frac {\log \left (3 e^5+2 x\right )}{3 e^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*E^5)/(3*E^5*x + 2*x^2),x]

[Out]

-9*E^5*(Log[x]/(3*E^5) - Log[3*E^5 + 2*x]/(3*E^5))

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fricas [A]  time = 0.69, size = 16, normalized size = 0.84 \begin {gather*} 3 \, \log \left (2 \, x + 3 \, e^{5}\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(5)/(3*x*exp(5)+2*x^2),x, algorithm="fricas")

[Out]

3*log(2*x + 3*e^5) - 3*log(x)

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giac [B]  time = 0.19, size = 25, normalized size = 1.32 \begin {gather*} 3 \, {\left (e^{\left (-5\right )} \log \left ({\left | 2 \, x + 3 \, e^{5} \right |}\right ) - e^{\left (-5\right )} \log \left ({\left | x \right |}\right )\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(5)/(3*x*exp(5)+2*x^2),x, algorithm="giac")

[Out]

3*(e^(-5)*log(abs(2*x + 3*e^5)) - e^(-5)*log(abs(x)))*e^5

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maple [A]  time = 0.10, size = 17, normalized size = 0.89

method	result	size
norman	\(-3 \ln \relax (x )+3 \ln \left (3 \,{\mathrm e}^{5}+2 x \right )\)	\(17\)
risch	\(-3 \ln \relax (x )+3 \ln \left (3 \,{\mathrm e}^{5}+2 x \right )\)	\(17\)
meijerg	\(3 \ln \left (1+\frac {2 x \,{\mathrm e}^{-5}}{3}\right )-3 \ln \relax (x )-3 \ln \relax (2)+3 \ln \relax (3)+15\)	\(25\)
default	\(-9 \,{\mathrm e}^{5} \left (\frac {\ln \relax (x ) {\mathrm e}^{-5}}{3}-\frac {{\mathrm e}^{-5} \ln \left (3 \,{\mathrm e}^{5}+2 x \right )}{3}\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-9*exp(5)/(3*x*exp(5)+2*x^2),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)+3*ln(3*exp(5)+2*x)

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maxima [A]  time = 0.37, size = 23, normalized size = 1.21 \begin {gather*} 3 \, {\left (e^{\left (-5\right )} \log \left (2 \, x + 3 \, e^{5}\right ) - e^{\left (-5\right )} \log \relax (x)\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(5)/(3*x*exp(5)+2*x^2),x, algorithm="maxima")

[Out]

3*(e^(-5)*log(2*x + 3*e^5) - e^(-5)*log(x))*e^5

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mupad [B]  time = 2.38, size = 10, normalized size = 0.53 \begin {gather*} 6\,\mathrm {atanh}\left (\frac {4\,x\,{\mathrm {e}}^{-5}}{3}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*exp(5))/(3*x*exp(5) + 2*x^2),x)

[Out]

6*atanh((4*x*exp(-5))/3 + 1)

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sympy [A]  time = 0.18, size = 29, normalized size = 1.53 \begin {gather*} - 9 \left (\frac {\log {\relax (x )}}{3 e^{5}} - \frac {\log {\left (x + \frac {3 e^{5}}{2} \right )}}{3 e^{5}}\right ) e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-9*exp(5)/(3*x*exp(5)+2*x**2),x)

[Out]

-9*(exp(-5)*log(x)/3 - exp(-5)*log(x + 3*exp(5)/2)/3)*exp(5)

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